3.8: Surface Force Density

In many systems, the electric or magnetic force density is concentrated in a thin layer, usually comprising the interface between two regions. If the thickness of this layer is small compared to the dimensions of the adjacent regions and other lengths of interest, then the force per unit area on the interface may be used to describe the layer. An interfacial section is enclosed by the incremental volume of thickness \(\Lambda\) and area \(A = \delta_x \delta_y\), shown in Fig. 3.11.1. The surface force density is defined as a force per unit area of the interface in a limit in which first \(\Lambda\) and then \(A\) approach zero. The integration of the electric force density throughout the control volume is conveniently carried out using the appropriate stress tensor \(T_{ij}\) integrated over the enclosing surface. With n defined as the unit normal to the interface and \(\overrightarrow{i}_n\) the unit normal to the control surface, the surface force density is

\[ \overrightarrow{T} = \lim_{\Delta \to 0 \\ A \to 0} \frac{1}{A} \oint_S \overrightarrow{\overrightarrow{T}} \cdot \overrightarrow{i}_n \, da = [] \overrightarrow{\overrightarrow{T}} [] \cdot \overrightarrow{n} + \lim_{A \to 0} \frac{1}{A} \oint_C \int_{0^{-}}^{0^{+}} \overrightarrow{\overrightarrow{T}} \cdot \overrightarrow{i}_n\, d \nu\, dl \label{1} \]

Integration is divided into two parts. The first is the contribution from the surfaces external to the layer, having normals \(\overrightarrow{n}\) and \(-\overrightarrow{n}\), respectively. The second accounts for the "edges" of the volume wherethe surface cuts through the double layer. If fields within the layer are of the same order as those outside, contributions of the second integral vanish as \(\Delta \rightarrow 0\). In electroquasistatic systems, the double layer presents a case where the internal fields are sufficiently intense that the second term not only makes a.contribution but one that can dominate the first term. The remainder of this section is devoted to converting this contribution to a more useful form.

The distance normal to the interface is \(y\), with \(\mu, \xi\) orthogonal coordinates in the local inter-facial plane, as shown in Fig. 3.11.1. In the absence of a double layer, the electric field is of the same order of magnitude throughout, and hence in the limit \(\Delta \rightarrow 0\), the second term in Equation \ref{1} becomes negligible compared to the first. With the double layer, the stress contributions from the edges of the control volume are of the same order as those from the exterior surfaces.

As discussed in Sec. 2.10, the tangential electric field suffers a discontinuity through the double layer. However, the tangential field within the layer is of the same order as the external field. Because the thickness A over which the interior stresses act is much smaller than the linear dimensions \(\delta \xi\) and \(\delta \mu\), the internal stress contributions to the integrations around the periphery of the control volume are ignorable unless the double-layer charges are themselves responsible for a substantially larger internal field than external field. This double-layer-generated field is directed normal to the interface and dominates in determining the interior stresses. The stress taken now as represented by Eq. 3.7.19b of Table 3.10.1 is

\[ T_{ij} = E_i D_j - \delta_{ij} W^{'} \label{2} \]

where, in the case of a linearly polarized dielectric, the coenergy density W' is simply \(\varepsilon E^2/2\). Stress components associated with the dominant field in the double layer interior are essentially

\[ \begin{align} T_{\xi \xi} &\rightarrow -W^{'} \nonumber \\ T_{ij} &\rightarrow 0; \quad i \neq j \nonumber \end{align} \label{3} \]

The traction acting on the periphery of the control volume is therefore approximately

\[ \int_{0^{-}}^{0^{+}} T \cdot \overrightarrow{i}_n \, d\nu = - \int_{0^{-}}^{0^{+}} W^{'} \, d \nu \overrightarrow{i}_n \equiv \gamma_E \overrightarrow{i}_n \label{4} \]

0The normal vector In can be written as \(- \overrightarrow{n} \times \overrightarrow{d} l \), so that Equation \ref{1} becomes

\[ \overrightarrow{T} = [] \overrightarrow{\overrightarrow{T}} [] \cdot \overrightarrow{n} - \lim_{A \to 0} \frac{1}{A} \oint_C \gamma_E \overrightarrow{n} \times \overrightarrow{d}l \label{5} \]

In the limit \(A \overrightarrow 0\), the contour integral in Equation \ref{5} need only be evaluated to first order in \(\delta \xi \cdot \delta \mu\).Expansion about the origin, denoted by the subscript o, gives an approximate expression for the integral that becomes exact in the limit. The contour \(C\) is taken as rectangular with edges parallel to the \(\xi, \mu\) axes. The segment of length \(\delta \mu\) at \(\xi = \delta \xi /2\) has \(-\overrightarrow{n} \times \overrightarrow{d} l = \delta \mu (\overrightarrow{i}_{\xi} + \overrightarrow{n}_o \delta \xi/ R_1)\) and gives a contribution to the contour integral

\[ \bigg\{ [\gamma_E]_o + [\frac{\partial{\gamma_E}}{\partial{\xi}}]_o \frac{\delta \xi}{2} \bigg\} \bigg\{ \overrightarrow{i}_{\xi} + \frac{\overrightarrow{n}_o \delta \xi}{R_1} \bigg\} \delta \mu \label{6} \]

The three additional sides of the rectangular contour give similar contributions, so that altogether,

\[ \begin{align} &- \lim_{A \to 0} \frac{1}{A} \oint_C \gamma_E \overrightarrow{n} \times \overrightarrow{d} l = \lim_{\delta \xi \delta \mu \to 0} \frac{1}{\delta \xi \delta \mu} \Bigg\{ \bigg\{ [\gamma_E]_o + [\frac{\partial{\gamma_E}}{\partial{\xi}}]_o \frac{\delta \xi}{2} \bigg\} \bigg\{ \overrightarrow{i}_{\xi} + \frac{\overrightarrow{n}_o}{R_1} \frac{\delta \xi}{2} \bigg\} \delta \mu \nonumber \\ &+ \bigg\{ [\gamma_E]_o - [\frac{\partial{\gamma_E}}{\partial{\xi}}]_o \frac{\delta \xi}{2} \bigg\} \bigg\{ -\overrightarrow{i}_{\xi} + \frac{\overrightarrow{n}_o}{R_1} \frac{\delta \xi}{2} \bigg\} \delta \mu + \bigg\{ [\gamma_E]_o + [\frac{\partial{\gamma_E}}{\partial{\mu}}]_o \frac{\delta \mu}{2} \bigg\} \bigg\{\overrightarrow{i}_{\mu} + \frac{\overrightarrow{n}_o}{R_2} \frac{\delta \mu}{2} \bigg\} \delta \xi \nonumber \\ & \bigg\{ [\gamma_E]_o - [-\frac{\partial{\gamma_E}}{\partial{\mu}}]_o \frac{\delta \mu}{2} \bigg\} \bigg\{-\overrightarrow{i}_{\mu} + \frac{\overrightarrow{n}_o}{R_2} \frac{\delta \mu}{2} \bigg\} \delta \xi \nonumber \Bigg\} \nonumber \\ &= \overrightarrow{n} \gamma_E [\frac{1}{R_1} + \frac{1}{R_2}] + \Delta_{\Sigma} \gamma_E \label{7} \end{align} \]

Here, \(R_1\) and \(R_2\) are radii of curvature for the interface, reckoned in the orthogonal planes defined respectively by the normal and \(\xi\) and the normal and \(\mu\). Note that the sign of each curvature term is taken as positive if the center of curvature is on the side of the interface toward which \(\overrightarrow{n}\) is directed. The surface force density associated with surface tension takes this same form. However,the convention used in Chapter 7 is with the radii of curvature the negatives of \(R_1\) and \(R_2\).With the understanding that \(R_1\) and \(R_2\) are radii of curvature taken as positive if the center of curvature is on the side of the interface out of which \(\overrightarrow{n}\) is directed, Eqs. \ref{1}, \ref{4}, and \ref{7} give the surface force density,with the double-layer contribution represented by the function \(\gamma_E\),

\[ \overrightarrow{T} = [] \overrightarrow{\overrightarrow{T}} \cdot \overrightarrow{n} - \overrightarrow{n} \gamma_E [\frac{1}{R_1} + \frac{1}{R_2}] + \Delta_{\Sigma} \gamma_E \label{8} \]

where

\[ \gamma_E \equiv \int_{0^{-}}^{0^{+}} W^{'} d \nu \nonumber \]

It is shown in Sec. 7.6 that the second term in Equation \ref{8} can also be expressed as \(- \gamma_E (\Delta \cdot \overrightarrow{n}) \overrightarrow{n}\).

The double layer surface force density is exemplified in Chapter 10.